May  2012, 6(2): 121-130. doi: 10.3934/amc.2012.6.121

On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$

1. 

Department of Mathematics, KTH, S-100 44 Stockholm

2. 

Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli, 1, I-06123 Perugia

Received  January 2011 Revised  August 2011 Published  April 2012

It is proved that for every integer $n=2^k-1$, with $k\geq5$, there exists a perfect code $C$ of length $n$, of rank $r=n-\log(n+1)+2$ and with a trivial symmetry group. This result extends an earlier result by the authors that says that for any length $n=2^k-1$, with $k\geq5$, and any rank $r$, with $n-\log(n+1)+3\leq r\leq n-1$ there exist perfect codes with a trivial symmetry group.
Citation: Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$. Advances in Mathematics of Communications, 2012, 6 (2) : 121-130. doi: 10.3934/amc.2012.6.121
References:
[1]

S. V. Avgustinovich, O. Heden and F. I. Solov'eva, The classification of some perfect codes,, Des. Codes Cryptogr., 31 (2004), 313.  doi: 10.1023/B:DESI.0000015891.01562.c1.  Google Scholar

[2]

S. V. Avgustinovich, O. Heden and F. I. Solov'eva, On the structure of symmetry groups of Vasilev codes,, Probl. Inform. Transm., 41 (2005), 105.  doi: 10.1007/s11122-005-0015-5.  Google Scholar

[3]

O. Heden, On the kernel of binary perfect 1-error correcting codes of length 15,, manuscript, (1987).   Google Scholar

[4]

O. Heden, A survey of perfect codes,, Adv. Math. Commun., 2 (2008), 223.  doi: 10.3934/amc.2008.2.223.  Google Scholar

[5]

O. Heden, F. Pasticci and T. Westerbäck, On the existence of extended perfect binary codes with trivial symmetry group,, Adv. Math. Commun., 3 (2009), 295.  doi: 10.3934/amc.2009.3.295.  Google Scholar

[6]

K. T. Phelps, A general product construction for error correcting Codes,, SIAM J. Algebra Discrete Methods, 5 (1984), 224.  doi: 10.1137/0605023.  Google Scholar

[7]

K. T. Phelps, O. Pottonen and P. R. J. Östergård, The perfect binary one-error-correcting codes of length 15: Part II properties,, IEEE Trans. Inform. Theory, 56 (2010), 2571.  doi: 10.1109/TIT.2010.2046197.  Google Scholar

[8]

F. I. Solov'eva, "On Perfect Codes and Related Topics,'', Pohang, (2004).   Google Scholar

[9]

V. A. Zinoviev, On generalized concatenated codes,, in, (1975), 587.   Google Scholar

[10]

V. A. Zinoviev, Generalized cascade codes,, Probl. Inform. Transm., 12 (1976), 5.   Google Scholar

[11]

V. A. Zinoviev and D. A. Zinoviev, Binary perfect and extended perfect codes of length 15 and 16 with ranks 13 and 14 (in Russian),, Problemy Peredachi Informatsii, 46 (2010), 20.   Google Scholar

show all references

References:
[1]

S. V. Avgustinovich, O. Heden and F. I. Solov'eva, The classification of some perfect codes,, Des. Codes Cryptogr., 31 (2004), 313.  doi: 10.1023/B:DESI.0000015891.01562.c1.  Google Scholar

[2]

S. V. Avgustinovich, O. Heden and F. I. Solov'eva, On the structure of symmetry groups of Vasilev codes,, Probl. Inform. Transm., 41 (2005), 105.  doi: 10.1007/s11122-005-0015-5.  Google Scholar

[3]

O. Heden, On the kernel of binary perfect 1-error correcting codes of length 15,, manuscript, (1987).   Google Scholar

[4]

O. Heden, A survey of perfect codes,, Adv. Math. Commun., 2 (2008), 223.  doi: 10.3934/amc.2008.2.223.  Google Scholar

[5]

O. Heden, F. Pasticci and T. Westerbäck, On the existence of extended perfect binary codes with trivial symmetry group,, Adv. Math. Commun., 3 (2009), 295.  doi: 10.3934/amc.2009.3.295.  Google Scholar

[6]

K. T. Phelps, A general product construction for error correcting Codes,, SIAM J. Algebra Discrete Methods, 5 (1984), 224.  doi: 10.1137/0605023.  Google Scholar

[7]

K. T. Phelps, O. Pottonen and P. R. J. Östergård, The perfect binary one-error-correcting codes of length 15: Part II properties,, IEEE Trans. Inform. Theory, 56 (2010), 2571.  doi: 10.1109/TIT.2010.2046197.  Google Scholar

[8]

F. I. Solov'eva, "On Perfect Codes and Related Topics,'', Pohang, (2004).   Google Scholar

[9]

V. A. Zinoviev, On generalized concatenated codes,, in, (1975), 587.   Google Scholar

[10]

V. A. Zinoviev, Generalized cascade codes,, Probl. Inform. Transm., 12 (1976), 5.   Google Scholar

[11]

V. A. Zinoviev and D. A. Zinoviev, Binary perfect and extended perfect codes of length 15 and 16 with ranks 13 and 14 (in Russian),, Problemy Peredachi Informatsii, 46 (2010), 20.   Google Scholar

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